reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th53:
  arctan|[.-1,1.] is continuous
proof
  set f = tan | [.-PI/4,PI/4.];
A1: f|[.-PI/4,PI/4.] = f by RELAT_1:72;
  (f|[.-PI/4,PI/4.])"|(f.:[.-PI/4,PI/4.]) is continuous by Lm11,Lm13,FCONT_1:47
;
  then arctan | [.-1,1.]|[.-1,1.] is continuous by A1,Th21,Th25,RELAT_1:115;
  hence thesis by FCONT_1:15;
end;
